Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $p = \dfrac{30q + 36}{8q} \div \dfrac{10q + 12}{-10} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{30q + 36}{8q} \times \dfrac{-10}{10q + 12} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ (30q + 36) \times -10 } { 8q \times (10q + 12) } $ $ p = \dfrac {-10 \times 6(5q + 6)} {8q \times 2(5q + 6)} $ $ p = \dfrac{-60(5q + 6)}{16q(5q + 6)} $ We can cancel the $5q + 6$ so long as $5q + 6 \neq 0$ Therefore $q \neq -\dfrac{6}{5}$ $p = \dfrac{-60 \cancel{(5q + 6})}{16q \cancel{(5q + 6)}} = -\dfrac{60}{16q} = -\dfrac{15}{4q} $